Reciprocal theorem for the prediction of the normal force induced on a particle translating parallel to an elastic membrane

When an elastic object is dragged through a viscous fluid tangent to a rigid boundary, it experiences a lift force perpendicular to its direction of motion. An analogous lift occurs when a rigid symmetric object translates parallel to an elastic interface or a soft substrate. The induced lift force is attributed to an elastohydrodynamic coupling that arises from the breaking of the flow reversal symmetry produced by the elastic deformation of the translating object or the interface. Here we derive explicit analytical expressions for the quasi-steady-state lift force exerted on a rigid spherical particle translating parallel to a finite-sized membrane exhibiting a resistance toward both shear and bending. Our analytical approach applies the Lorentz reciprocal theorem so as to obtain the solution of the flow problem using a perturbation technique for small deformations of the membrane. We find that the shear-related contribution to the normal force leads to an attractive interaction between the particle and the membrane. This emerging attractive force decreases quadratically with the system size to eventually vanish in the limit of an infinitely extended membrane. In contrast, membrane bending leads to a repulsive interaction whose effect becomes more pronounced upon increasing the system size, where the lift force is found to diverge logarithmically for an infinitely large membrane. The unphysical divergence of the bending-induced lift force can be rendered finite by regularizing the solution with a cutoff length beyond which the bending forces become subdominant to an external body force.

Figure 1

Illustration of the elastohydrodynamic problem. A solid sphere of radius $a$ located a distance $h$ above an elastic membrane of radius $b$. In the undeformed state, the membrane is extended in the plane $z=0$. The frame of reference attached to the center of the sphere translates at a constant velocity ${\mathit{V}}_{\mathrm{P}}$ with respect to the laboratory frame. The fluid on both sides of the membrane has the same dynamic viscosity $\eta$. The figure in the inset is a top view of the frame of reference associated with the particle where $(r,\varphi )$ are the polar coordinates.

Figure 2

Variation of the rescaled normal lift force due to particle motion tangent to a finite-sized membrane with (a) pure bending and (b) pure shear versus the parameter $\lambda ={\left(1+b^2\right)}^{1/2}$ as predicted theoretically from Eqs. (21) and (22). The blue dashed line shown in panel (a) is the asymptotic results given by Eq. (23a).

Figure 3

Variation of the rescaled normal lift force due to an infinitely extended membrane with pure bending vs ${\epsilon }^{-1}$ as given by Eq. (33). The dashed line is an asymptotic fit to the lift force in the region ${\epsilon }^{-1}\gg 1$. The inset shows the same data in a log-log plot where an asymptotic fit in the region ${\epsilon }^{-1}\ll 1$ is shown as a dashed line.

Figure 4

Rescaled membrane displacement in the plane of maximum deformation for (a) and (b)$\varphi =0$, and (c)$\varphi =\pi /4$, as predicted theoretically for a membrane size $b^{★}=20$ and Skalak ratio $C=1$.